Curve skeleton extraction from incomplete point cloud

Abstract

We present an algorithm for curve skeleton extraction from imperfect point clouds where large portions of the data may be missing. Our construction is primarily based on a novel notion of generalized rotational symmetry axis (ROSA) of an oriented point set. Specifically, given a subset S of oriented points, we introduce a variational definition for an oriented point that is most rotationally symmetric with respect to S. Our formulation effectively utilizes normal information to compensate for the missing data and leads to robust curve skeleton computation over regions of a shape that are generally cylindrical. We present an iterative algorithm via planar cuts to compute the ROSA of a point cloud. This is complemented by special handling of non-cylindrical joint regions to obtain a centered, topologically clean, and complete 1D skeleton. We demonstrate that quality curve skeletons can be extracted from a variety of shapes captured by incomplete point clouds. Finally, we show how our algorithm assists in shape completion under these challenges by developing a skeleton-driven point cloud completion scheme.

References

1. Amenta, N., and Kil, Y. J. 2004. Defining point-set surfaces. ACM Trans. on Graph. 23, 3, 264--270.
2. Au, O. K.-C., Tai, C.-L., Chu, H.-K., Cohen-Or, D., and Lee, T.-Y. 2008. Skeleton extraction by mesh contraction. ACM Trans. on Graph. 27, 3, 44:1--44:10.
3. Blum, H. 1967. A transformation for extracting new descriptors of shape. Models for the perception of speech and visual form. MIT Press, 362--380.
4. Bouix, S., Siddiqi, K., Tannenbaum, A., and Zucker, S. 2006. Medial axis computation and evolution. Statistics and Analysis of Shapes.
5. Carr, J. C., Beatson, R. K., Cherrie, J. B., Mitchell, T. J., Fright, W. R., McCallum, B. C., and Evans, T. R. 2001. Reconstruction and representation of 3D objects with radial basis functions. In Proc. of SIGGRAPH, 67--76.
6. Chuang, J.-H., Ahuja, N., Lin, C.-C., Tsai, C.-H., and Chen, C.-H. 2004. A potential-based generalized cylinder representation. Computers & Graphics 28, 6, 907--918.
7. Cornea, N. D., Min, P., and Silver, D. 2007. Curve-skeleton properties, applications, and algorithms. IEEE Trans. Vis. & Comp. Graphics 13, 3, 530--548.
8. de Aguiar, E., Theobalt, C., Thrun, S., and Seidel, H.-P. 2008. Automatic conversion of mesh animations into skeleton-based animations. Computer Graphics Forum (Proc. of Eurographics) 27, 2 (4), 389--397.
9. Dey, T., and Sun, J. 2006. Defining and computing curve-skeletons with medial geodesic function. In Symp. on Geom. Proc., 143--152.
10. Giblin, P. J., and Brassett, S. A. 1985. Local symmetry of plane curves. American Mathematical Monthly, 689--707.
11. Hilaga, M., Shinagawa, Y., Kohmura, T., and Kunii, T. L. 2001. Topology matching for fully automatic similarity estimation of 3D shapes. In Proc. of SIGGRAPH, 203--212.
12. Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., and Stuetzle, W. 1992. Surface reconstruction from unorganized points. In Proc. of SIGGRAPH, 71--78.
13. James, D., and Twigg, C. 2005. Skinning mesh animations. ACM Trans. on Graph. 24, 3, 399--407.
14. Katz, S., and Tal, A. 2003. Hierarchical mesh decomposition using fuzzy clustering and cuts. ACM Trans. on Graph. 22, 3, 954--961.
15. Kazhdan, M., Funkhouser, T., and Rusinkiewicz, S. 2004. Symmetry descriptors and 3D shape matching. In Symp. on Geom. Proc., 115--123.
16. Kazhdan, M., Bolitho, M., and Hoppe, H. 2006. Poisson surface reconstruction. In Symp. on Geom. Proc., 61--70.
17. Kim, Y. J., Varadhan, G., Lin, M. C., and Manocha, D. 2003. Fast swept volume approximation of complex polyhedral models. In Proc. of Symp. on Solid Modeling and App., 11--22.
18. Lee, I. 2000. Curve reconstruction from unorganized points. Computer Aided Geometric Design 17, 2, 161--177.
19. Lehtinen, J., Zwicker, M., Turquin, E., Kontkanen, J., Durand, F., Sillion, F., and Aila, T. 2008. A meshless hierarchical representation for light transport. ACM Trans. on Graph. 27, 3, 37:1--37:10.
20. Lewis, J. P., Cordner, M., and Fong, N. 2000. Pose space deformation: a unified approach to shape interpolation and skeleton-driven deformation. In Proc. of SIGGRAPH, 165--172.
21. Li, X., Woon, T., Tan, T., and Huang, Z. 2001. Decomposing polygon meshes for interactive applications. In Proc. of Symposium on Interactive 3D graphics, 35--42.
22. Lipman, Y., Cohen-Or, D., Levin, D., and Tal-Ezer, H. 2007. Parameterization-free projection for geometry reconstruction. ACM Trans. on Graph. 26(3), 22:1--22:6.
23. Lu, L., Choi, Y.-K., Wang, W., and Kim, M.-S. 2007. Variational 3D shape segmentation for bounding volume computation. Computer Graphics Forum (Proc. of Eurographics) 26, 3, 329--338.
24. Malandain, G., and Fernández-Vidal, S. 1998. Euclidean skeletons. Image and Vision Computing 16, 5, 317--327.
25. Mitra, N., Guibas, L., and Pauly, M. 2006. Partial and approximate symmetry detection for 3D geometry. ACM Trans. on Graph. 25, 3, 560--568.
26. Mitra, N. J., Flöry, S., Ovsjanikov, M., Gelfand, N., Guibas, L., and Pottmann, H. 2007. Dynamic geometry registration. In Symp. on Geom. Proc., 173--182.
27. Nehab, D., Rusinkiewicz, S., Davis, J., and Ramamoorthi, R. 2005. Efficiently combining positions and normals for precise 3D geometry. ACM Trans. on Graph. 24, 3, 536--543.
28. Ogniewicz, R., Ilg, M., and Zurich, E. 1992. Voronoi skeletons: theory and applications. In Proc. IEEE Conf. on Comp. Vis. and Pat. Rec., 63--69.
29. Ovsjanikov, M., Sun, J., and Guibas, L. 2008. Global intrinsic symmetries of shapes. In Computer Graphics Forum (Proc. of Symp. on Geom. Proc.), vol. 27, 1341--1348.
30. Patane, G., Spagnuolo, M., and Falcidieno, B. 2008. Reeb graph computation based on a minimal contouring. In Proc. IEEE Conf. on Shape Modeling and App., 73--82.
31. Pekelny, Y., and Gotsman, C. 2008. Articulated object reconstruction and motion capture from depth video. Computer Graphics Forum (Proc. of Eurographics) 27, 2, 399--408.
32. Podolak, J., Shilane, P., Golovinskiy, A., Rusinkiewicz, S., and Funkhouser, T. 2006. A planar-reflective symmetry transform for 3D shapes. ACM Trans. on Graph. 25, 3, 549--559.
33. Raab, R., Gotsman, C., and Sheffer, A. 2004. Virtual woodwork: Making toys from geometric models. Int. J. of Shape Modeling 10, 1, 1--30.
34. Sharf, A., Lewiner, T., Shamir, A., and Kobbelt, L. 2007. On-the-fly curve-skeleton computation for 3D shapes. Computer Graphics Forum (Proc. of Eurographics) 26, 3, 323--328.
35. Sharf, A., Lewiner, T., Shklarski, G., Toledo, S., and Cohen-Or, D. 2007. Interactive topology-aware surface reconstruction. ACM Trans. on Graph. 26, 3, 43:1--43:10.
36. Sharf, A., Alcantara, D. A., Lewiner, T., Greif, C., Sheffer, A., Amenta, N., and Cohen-Or, D. 2008. Space-time surface reconstruction using incompressible flow. ACM Trans. on Graph. 27, 5, 110:1--110:10.
37. Siddiqi, K., and Pizer, S. 2009. Medial Representations: Mathematics, Algorithms and Applications. Springer.
38. Simari, P. D., and Singh, K. 2005. Extraction and remeshing of ellipsoidal representations from mesh data. In Proc. of Graphics Interface, 161--168.
39. Sorkine, O., and Cohen-Or, D. 2004. Least-squares meshes. In Proc. IEEE Conf. on Shape Modeling and App., 191--199.
40. Thrun, S., and Wegbreit, B. 2005. Shape from symmetry. In Proc. of Int. Conf. on Comp. Vis., vol. 2, 1824--1831.
41. Wand, M., Jenke, P., Huang, Q., Bokeloh, M., Guibas, L., and Schilling, A. 2007. Reconstruction of deforming geometry from time-varying point clouds. In Symp. on Geom. Proc., 49--58.

Index Terms

• Computing methodologies

  • Artificial intelligence

    • Computer vision

      • Computer vision problems

        • Interest point and salient region detections

  • Computer graphics

    • Image manipulation

    • Shape modeling

      • Parametric curve and surface models

      • Volumetric models

  • Machine learning

    • Machine learning algorithms

      • Feature selection

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry